In sec. 2.2.4 we have presented several different realizations of allpass filtersbecause they find many applications in signal processing [76]. In particular, acouple of allpass filters is often combined in a parallel structure in such a waythat the overall response is not allpass. If H

a1

and H

a2

are two different allpass

filters, their parallel connection, having transfer function H

l

(z) = H

a1

(z) +

H

a2

(z) is not allpass. To figure this out, just think about frequencies where the

two phase responses are equal. At these points the signal will be doubled atthe output of H(z). On the other hand, at points where the phase response aredifferent by (i.e., they are in phase opposition), the outputs of the two branchescancel out at the output. In order to design a lowpass filter it is sufficient toconnect in parallel two allpass filters having a phase response similar to that offig. 29.The same parallel connection, with a subtraction instead of the additionat the output, gives rise to a highpass filter H

h

(z), and it is possible to show

that the highpass and the lowpass transfer functions are complementary, in thesense that |H

l

()|

2

+ |H

h

()|

2

is constant in frequency. Therefore, we have the

a

Figure 29: Phase responses of two allpass filters that, if connected in parallel,give a lowpass filter

compact realization of a crossover filter, as depicted in fig. 30, which is a devicewith one input and two outputs that conveys the low frequencies to one outlet,and the high frequencies to the other outlet. Devices such as this are found notonly in loudspeakers, but also in musical instrument models. For instance, thebell of woodwinds transmits to the air the high frequencies and reflects the lowfrequencies back to the bore.

H (z)

H (z)

a1

a2

-1

1/2

x

y

y

1

2

Figure 30: Crossover implemented as a parallel of allpass filters and a latticejunction

The idea of connecting two allpass filters in parallel can be applied to the

realization of resonant complementary filters. In particular, it is interesting to